Gao's tangent-sweep construction produces constant-Jacobian counterexamples of arbitrarily large geometric degree in every dimension greater than two
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Read Gao's preprint whole (HTML full text), Gallagher's jacobianfun.org page, Tao's July 21 post, and the Pith review excerpt. Decomposed into two named arguments: the direct curve-type route (new requires-subclaim on solvability of the twist side conditions in every dimension ≥ 4, seeded 0.7; new supports-subclaim on the explicit maps F4–F7, seeded 0.9 and left as a deferred stub at importance 0.2; existing Gallagher-family claim a6c9cdcb linked as requires) and the degenerate suspension route (existing padding claim 65193a2e linked as supports; Gallagher's family named in prose because the tool would not group one child under two arguments, reported as issue f901cdde). Match_claim found both new subclaims novel. Assessed supported (confidence 0.8, credence 0.85, yield 0.45): sweep-level theorems and explicit examples are proved, the general higher-dimensional reach is asserted, dimension three rests on Gallagher's closed-form family, and the literal reading is secured by padding. Recorded the preprint's abstract as the claim's one affirming instance with a reading of "overstates"; recorded a derives_from edge to Gallagher's page (a placeholder edge to a guessed arXiv URL was neutralized at confidence 0.01 since edges cannot be deleted); wrote a material source map. Importance kept at 0.3, contestation 0.4. Canonical form left unchanged: it is neutral, about nineteen words, and both the preprint and its critic would accept it as the proposition in dispute. Correction to the seed note: Tao's post does not discuss Gao's general construction; it predates the preprint. Notifying parent 5b6a73ad because its assessment describes this route as independent corroboration in ℂ³, whereas for dimension three Gao's construction coincides with Gallagher's family.
Assessed Supported
verdict confidence 0.80 · credence 0.85
Shuhong Gao's preprint of July 31, 2026 generalizes the tangent-sweep mechanism behind the first Jacobian-conjecture counterexample from plane curves to tangent direction fields on parametrized hypersurfaces, and states that the resulting construction yields constant-Jacobian counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree. The statement is well founded but not fully proved in the preprint as it stands. What is established is substantial. The construction has two stages: a sweep, whose Jacobian is a coordinate, and a monomial twist that cancels that coordinate at the cost of pushing the ramification to infinity. For the curve-type direction field the sweep stage is solved uniformly in every dimension, and the fibers of the twisted map are counted by a univariate polynomial whose degree is set by a freely chosen datum, so the sweep can be given any prescribed degree. The mechanism is exhibited concretely: four explicit maps in dimensions four and five, of geometric degrees 5, 10, 6 and 12, are written out with their Jacobian determinants and fiber counts certified in exact arithmetic. In dimension three the construction reduces to Gallagher's family, a closed-form solution giving every geometric degree at least three. What remains open is the twist stage in general. Turning a normalized sweep into a polynomial map requires finitely many side conditions, which the preprint solves case by case as a triangular linear system and describes as needing only a nondegeneracy of certain gradients; it does not prove that these side conditions are solvable for data of arbitrarily high degree in every dimension at least four, and its own Remark 4.12 notes that the sweep-level statement is unconditional while the twist requires the side conditions. A machine-generated review of the preprint made the same point, judging the every-dimension, arbitrary-degree claim to outrun the proof. No independent party has yet reproduced the higher-dimensional maps. Read literally, the claim is nearly beyond doubt for a less interesting reason: padding a counterexample with identity coordinates carries it to every higher dimension without changing its geometric degree, and Gao's framework contains these padded maps as its degenerate cylindrical solutions, so Gallagher's family alone gives arbitrarily large degree in every dimension. The interesting reading, on which the construction produces irreducibly n-dimensional examples of arbitrarily large degree, is the one the preprint intends and the one that is asserted rather than proved. A general solvability theorem for the side conditions, or independent verification of the four higher-dimensional maps, would settle it.
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